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Widely-used Android static program analysis tools, e.g., Amandroid and FlowDroid, perform the whole-app inter-procedural analysis that is comprehensive but fundamentally difficult to handle modern (large) apps. The average app size has increased three to four times over five years. In this paper, we explore a new paradigm of targeted inter-procedural analysis that can skip irrelevant code and focus only on the flows of security-sensitive sink APIs. To this end, we propose a technique called on-the-fly bytecode search, which searches the disassembled app bytecode text just in time when a caller needs to be located. In this way, it guides targeted (and backward) inter-procedural analysis step by step until reaching entry points, without relying on a whole-app graph. Such search-based inter-procedural analysis, however, is challenging due to Java polymorphism, callbacks, asynchronous flows, static initializers, and inter-component communication in Android apps. We overcome these unique obstacles in our context by proposing a set of bytecode search mechanisms that utilize flexible searches and forward object taint analysis. Atop of this new inter-procedural analysis, we further adjust the traditional backward slicing and forward constant propagation to provide the complete dataflow tracking of sink API calls. We have implemented a prototype called BackDroid and compared it with Amandroid in analyzing 3,178 modern popular apps for crypto and SSL misconfigurations. The evaluation shows that for such sink-based problems, BackDroid is 37 times faster (2.13 v.s. 78.15 minutes) and has no timed-out failure (v.s. 35% in Amandroid), while maintaining close or even better detection effectiveness.